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**Analysis of failure occurrence from direct simulations**

Luc Sibille, François Nicot, Frédéric-Victor Donzé, Félix Darve

**To cite this version:**

Luc Sibille, François Nicot, Frédéric-Victor Donzé, Félix Darve. Analysis of failure occurrence from direct simulations. European Journal of Environmental and Civil Engineering, Taylor & Francis, 2009, 13 (2), pp.187-201. �10.1080/19648189.2009.9693099�. �hal-01005248�

**Analysis of failure occurrence **

**from direct simulations **

**Luc Sibille* — François Nicot** — Frédéric-Victor Donzé*** **

**Félix Darve*** **

* *Laboratoire GeM, Université de Nantes, ECN, CNRS*

*IUT de Saint-Nazaire, BP 420, F-44606 Saint-Nazaire cedex *
*luc.sibille@univ-nantes.fr *

*** Cemagref de Grenoble, Unité ETNA *
*BP 76, F-38402 Saint-Martin-d’Hères cedex *
*francois.nicot@cemagref.fr *

**** Laboratoire 3S-R, INPG, UJF, CNRS *
*BP 53, F-38041 Grenoble cedex 9 *

*{Frederic.Donze, Felix.Darve}@inpg.fr *

*ABSTRACT.Stress probes are simulated with the discrete element method (DEM). From these *

*simulations we show that the numerical discrete model presents a non-associative flow rule. *
*As the material is non-associate, sign of the second-order work is checked for stress states *
*included within the plastic limit condition. Conditions of occurrence of failure when the *
*second-order work vanishes are discussed. *

*RÉSUMÉ.* *Des recherches directionnelles en contrainte sont simulées avec la méthode des *
*éléments discrets (MED). A partir de ces simulations, nous montrons que le modèle *
*numérique discret présente une règle d’écoulement non associée. Par conséquent, le signe du *
*travail du second ordre est calculé pour des états de contraintes inclus à l’intérieur de la *
*condition limite de plasticité. Les conditions de développement d’une rupture lorsque le *
*travail du second ordre s’annule sont discutées. *

*KEYWORDS:* *failure, second-order work, control parameter, flow rule, stress probes, discrete *
*element method. *

*MOTS-CLÉS :* *rupture, travail du second ordre, paramètre de contrôle, règle d’écoulement, *

**1. Introduction**

Classically in geomechanics, failure phenomenon is associated with the existence
of limit stress states. Experimental observations show that these limit stress states
cannot be exceeded. If an operator tries to impose an additional stress loading
in-crement from a limit stress state, the strain response will be very “large” (or even
“unbounded”), the material fails. Notions of limit state and failure have been
*gener-alized by Darve et al. (2004) for stress states strictly included within the plastic limit*
condition. In the framework of rate-independent materials, generalized limit states
are interpreted as homogeneous bifurcation states: according to the control
parame-ters chosen, different responses are possible from the bifurcation state, some of the
responses could correspond to the failure of the material. These bifurcation states (or
equivalently, mixed limit states) are detected when the second-order work is zero or
negative:

d2W = dσ

∼ : dε∼ ≤ 0 [1]

*As shown by Darve et al. (2004), d*2W is essentially a directional quantity. In
other words, a value of d2W does not correspond to a given mechanical state, but
corresponds to a given loading direction (in stress or strain space) from a given
stress-strain state. In addition, d2W can vanish inside the plastic limit condition only for
non-associated materials, for associated materials d2W vanishes on the plastic limit
*condition. Darve et al. (2004) have shown that a whole bifurcation domain in stress*
space (including stress states for which d2W ≤ 0 for one or several stress directions)
exist for the Hostun sand.

*From different basis and approaches, Nicot et al. (2007), with the notion of *
*“sus-tainability”, and Nova (1994) (or Imposimato et al. (1998)), with the notion of *
“con-trollability”, also discussed the occurrence of failure for stress states inside the plastic
limit condition. All these approaches are not detailed in this paper and we
encour-age readers to refer to the quoted papers. However these approaches converge toward
identical conditions to be fulﬁlled together, for the occurrence of failure: (i) stress state
belongs to the bifurcation domain; (ii) loading direction is characterised by d2W ≤ 0;
(iii) control parameters are the ones allowing to trigger the failure. These conditions
have been deduced from analytical analyses, and their necessity in failure occurrence
has not been systematically veriﬁed. Therefore, we present in this paper a numerical
veriﬁcation of these conditions, from direct simulations based on the discrete element
*method (Cundall et al., 1979). Non-associativeness of the numerical discrete model*
is ﬁrst discussed, since it is a necessity for vanishing of d2_{W} _{within the plastic limit}
condition.

**2. Numerical discrete model**

Direct simulations were performed on a numerical discrete model with the code
*SDEC developed by Donzé et al. (1997), and based on the discrete element method*

*Cundall et al. (1979). The 3D numerical model has a cubical shape, it is composed of*
about 10,000 polydisperse spheres (Figure 1a). Let’s denote d_{s} the sphere diameter,
then d_{s}max/d_{s}min = 4.75. Spheres are rigid, they can slightly overlap according to an
interaction contact law presented in Figure 1b. In the direction normal to the tangent
contact plane, the relation is purely elastic and no tensile force is allowed. In the
direction included in the tangent contact plane, the relation is elastic perfectly plastic.
Only three mechanical parameters are introduced in the model at the contact scale;
the normal k_{n} and tangential k_{t} stiffnesses, and the friction angle ϕ_{c}. From sphere
positions and the interaction contact law, contact forces are computed at each time
step. Then, new sphere positions and orientations are determined by applying the
Newton’s second law from resultant forces and torques acting on each sphere. An
explicit scheme is used to integrate the Newton’s second law.

a)

*c*

*k*

*n*

*k*

*t*

b)

**Figure 1. a) numerical model with frictionless walls and frame axes; b) sketch of the***intergranular interaction law*

The macroscopic stress-strain state of the numerical model (or numerical sample)
is imposed through six frictionless walls whose positions are controlled at each time
step to follow the prescribed loading programme. Since there is no tangential forces
in wall-grain contacts, stress and strain principal directions coincide with the normals
to the walls (Figure 1a). Each principal value of stresses or strains can be controlled;
either directly for strains by adjusting the wall displacements, either indirectly for
stresses thanks to a closed-loop control (since only wall positions or displacements
are controlled). Consequently, all loading programmes deﬁned with principal values
of stresses (σ_{1}, σ_{2}, σ_{3}) and/or strains (ε_{1}, ε_{2}, ε_{3}) can be applied. The strain state is
determined from wall positions and the stress state from wall-grain contact forces.

In this paper, analyses are carried out from simulations performed on two
numeri-cal samples E1 and E3. The samples differ only in their initial density, their
character-istics are given in Table 1. During an axisymetric triaxial compression (characterised
by a constant radial stress), the densest sample E1 is dilatant, whereas the sample E3
is essentially contractant. All simulations presented hereafter are performed in
axisy-metric conditions (σ_{2} = σ_{3} and ε_{2} = ε_{3}), thus stress states and strain states can be
represented in the Rendulic (or axisymetric) planes of stresses (σ_{1}, √2 σ_{3}) or strains
(ε_{1}, √2 ε_{3}).

**Table 1. Characteristics of the numerical samples**

Sample kn/ds kt/kn ϕc Void ratio Coordination number

(MPa) (deg) e z

E1 356 0.42 35.0 0.618 4.54

E1 356 0.42 35.0 0.693 4.42

**3. Non associative ﬂow rule**

To investigate the incremental constitutive behaviour of the numerical samples,
it is useful to plot the response envelopes to strain or stress probes as suggested by
Gudehus (1979). Stress probes, deﬁned in the Rendulic plane of stress increments by
the norm dσ =(dσ_{1})2 + (dσ_{3})2 and the orientation angle α (see Figure 2a), are
performed from initial stress-strain states reached after drained triaxial compressions.
The initial stress states are characterised by the conﬁning pressure σ_{3} and the shear
stress ratio η = q/p (with q = σ_{1} _{− σ3} and p = (σ_{1} + 2σ_{3})/3). The norm of the
incremental stress loading * _{dσ is set equal to 1 kPa (i.e. of the order of 10}*−2 times
the mean pressure p imposed for the simulations). Since the response of the numerical
model to a given loading is non-linear, the choice of the norm of dσ is a tricky point.
dσ should be chosen as small as possible to limit the inﬂuence of the non-linearity of
the strain response on the shape of the response envelope. Nevertheless the response
of a discrete element model to a given loading is marked by successive quasi-static
phases followed by inertial phases (related to a modiﬁcation of the contact network)

*(Combe et al., 2000). Hence*

_{dσ should be large enough to include some of these}inertial phases (Sibille, 2006) directly involved in the macroscopic constitutive be-haviour of the numerical model. For each stress probe, orientation α is constant, and strain response is computed and characterised in Rendulic plane of strain increment by

_{dε =}(dε

_{1})2 + (dε

_{3})2 and the orientation β (see Figure 2b). The envelope of the set of strain responses dε computed for 0

_{≤ α ≤ 360 deg (by step of 10 deg)}from a given initial stress-strain state is called the response envelope. Such a response envelope is shown with crosses in Figure 3.

In the following, the computed response envelopes are discussed by making refer-ence to the classical elasto-plasticity framework. Thus, we assume:

– the classical decomposition of strains into an elastic part dε_{e} and a plastic part
dε_{p} such as:

dε = dε_{e} + dε_{p} [2]

– the existence of an elastic limit surface f, of normal n, separating an elastic tensorial zone and an elasto-plastic tensorial zone,

– the existence of a plastic potential g, of normal m.

To compute the elastic responses to stress probes, the sliding at intergranular contacts
is inhibited by imposing ϕ_{c} _{→ 90 deg. For the size of the stress increments imposed,}

this later condition is conﬁrmed to be sufﬁcient, by verifying the vanishing of residual strains after a cycle of loading/unloading, and by checking the lack of sliding and opening contacts. Once the elastic strain response is determined, the plastic strain response is computed from Equation [2].

a) de de3 de1 b b)

**Figure 2. Stress probes a) and strain responses b) defined in the Rendulic plane of***stress and strain increments, respectively*

−12 −10 −8 −6 −4 −2 0 2
x 10−5
−2
0
2
4
6
8
10
12x 10
−5
21/2 d_{ε}_{3}
dε 1
total strains
elastic strains
plastic strains

**Figure 3. Decomposition of the strain response envelope into an elastic part and a**

*plastic part; sample E1, σ*_{3} *= 200 kPa and η = 0.77*

In Figure 3 are presented a total response envelope, computed with sample E1, and
its decomposition into elastic and plastic response envelopes. The elastic response
en-velope with an elliptical shape centered with respect to the axis origin is typical of
a purely elastic behaviour (Gudehus, 1979). The plastic response envelope
(repre-sented with black points) forms a straight line meaning that the direction β_{p} of plastic
strain increments dε_{p} is constant and independent of stress loading directions α (for

a given initial stress-strain state). Therefore, this plastic response envelope shows the existence of a ﬂow rule for the numerical samples used.

0 90 180 270 360 0 0.2 0.4 0.6 0.8 1 1.2x 10 −4

Stress increment direction α (°)

|| d
ε p
|| or || d
ε e
||
plastic tensorial
zone
|| dε_{p} ||
|| dε_{e} ||
a)
0 90 180 270 360
−90
0
90
180
270
360

Stress increment direction α (°)

Elastic

β e

or plastic

β p

strain direction (°)

plastic tensorial zone

β_{p}≈ 129°

elastic strains plastic strain

b)

* Figure 4. a) norms of plastic,* dεp

*, and elastic, dε*e

*, strain responses; b)*

*direc-tions of plastic, β*_{p}*, and elastic, β*_{e}*, strains; sample E3, σ*_{3} *= 200 kPa and η = 0.63*

Another way to represent a strain response envelope consists in plotting the norm
and the direction of strain responses versus the stress loading direction. Figures 4a
and 4b display such representations for elastic and plastic strain response envelopes
computed with sample E3. Figure 4b shows clearly that β_{p} is quasi constant (for
dεp = 0) conﬁrming the existence of a ﬂow rule, and with βp = 129 deg
corre-sponding to the direction of the normal m to the plastic potential.

In Figure 5 are presented the stress directions corresponding to an elasto-plastic response in the sense of classical elasto-plasticity. By following these directions counter-clockwise, the ﬁrst and the last directions are tangent to the elastic limit sur-face f. Right and left tangents to the elastic limit sursur-face can be deduced from Fig-ure 4a: ﬁrst plastic strains are found for α = 60 deg and last ones for α = 240 deg.

Right and left tangents are collinear and the normal n to the elastic limit surface is
aligned along the direction α = 150 deg. This latter direction corresponds to the
max-imum ofdεp as shown in Figure 4a. By this way we obtain a rough estimation of the
direction of n, but this estimation is enough to conclude about the non-associativeness
*of the numerical model (see also Bardet (1994) and Calvetti et al. (2003)). *
There-fore the second-order work should vanish for stress states included in the plastic limit
surface.

**Figure 5. Definition of stress directions leading to an elasto-plastic response in the***framework of classical elasto-plasticity*

Figure 6 shows directions of n and m for different stress states and for samples E1 and E3. We remark for high value of η that the elastic limit surface tends to align with the Mohr-Coulomb criterion (plotted with a thin continuous line).

**4. Occurrence of diffuse failure**

Since the second-order work d2W is essentially a directional quantity, it is useful
to perform stress probes (as deﬁned in Section 3) to check values of d2_{W} _{with respect}
to stress loading directions. Values of second-order work are normalized to allow
comparisons between different stress-strain states and different sample densities. The
normalized second-order work d2W_{norm} *is deﬁned as (Darve et al., 2000):*

d2W_{norm} = dσ . dε

dσ dε [3]

d2W_{norm} is equal to the cosine of the angle between dσ and dε, consequently from
Figure 5 it appears clearly that d2_{W}_{norm} _{(and thus d}2_{W}_{) can take negative values}
before the plastic limit condition only if the material is non-associated.

0 100 200 300 400
0
100
200
300
400
500
*Sample E1*
21/2σ_{3} (kPa)
σ 1
(kPa)
*m*
*n*
η = 0.77
η = 0.31
η = 0.74
0 100 200 300 400
0
100
200
300
400
500
*Sample E3*
21/2σ_{3} (kPa)
σ 1
(kPa)
*m*
*n*
η = 0.63
η = 0.57

**Figure 6. Comparison of directions of the plastic flow, **m, and of the normal n to the

*elastic limit surface*

Figure 7 shows a circular diagram of d2W_{norm} versus the stress direction α
com-puted with samples E1 and E3 for a conﬁnement of 100 kPa. In a such diagram, an
arbitrary constant ρ is added to polar values of d2W_{norm} to verify:

∀α, d2_{W}

norm(α) + ρ > 0 [4]

A dashed circle drawn in Figure 7 represents vanishing values of d2W_{norm}. Inside
the dashed circle d2_{W}_{norm} _{is negative, outside it is positive. For both samples, we}
remark negative values of d2W_{norm} (or d2W) for stress directions α grouped in a
*cone of unstable stress directions (Imposimato et al., 2001; Darve et al., 2004). Cones*
shown here are the ﬁrst ones found with respect to the shear stress level η (η = 0.82
and 0.46 for E1 and E3 respectively, below these η values d2W > 0 for all
*stress-strain states and stress directions checked (Sibille et al., 2007; Sibille et al., 2008)).*
These shear stress levels (corresponding to mobilized friction angles φ_{m} = 21.0˚ and
12.3˚) are included within the Mohr-Coulomb criterion (deﬁned by φ = 24.7˚ and
21.2˚ for E1 and E3 respectively).

Through Figure 7 are summarized the three main characteristics of the inﬂuence
of density on cones of unstable stress directions. (i) the cone is more opened for the
loosest sample, (ii) for the loosest sample lower α values are included in the cone than
for the densest sample, (iii) ﬁrst cone is found for a lower value of φ_{m} with the loosest
sample than with the densest one. An identical qualitative inﬂuence of the density on
cones has been highlighted by computations made with a phenomenological
*consti-tutive relation ﬁtted on dense and loose Hostun sands (Darve et al., 2004; Sibille et*

*al., 2007). This simple comparison of cones of unstable stress directions with respect*

sam-ple from a stress state included within the Mohr-Coulomb criterion is much easier to observe than with a dense sample (the bifurcation domain and the range of unstable stress directions are wider).

30 210 60 240 90 270 120 300 150 330 180 0 sample E1; η = 0.82 sample E3; η = 0.46 q = cst. α = 215.3° d2W = 0

* Figure 7. Normalized second-order work, d*2W

_{norm}

*, versus stress probe direction α,*

*for samples E1 and E3 at the same confining pressure σ*_{3} *= 100 kPa*

For simulations of stress probes, numerical samples were fully stress controlled.
The loading programme was deﬁned for a given stress direction α by the control
pa-rameters dσ_{1} = cst_{1} and dσ_{3} = cst_{3} (plus dσ_{2} = dσ_{3} to verify the axisymetric
condition), where cst_{1} and cst_{3} are chosen such that the stress direction α is veriﬁed.
The corresponding response parameters are dε_{1} and dε_{3}. With these control
parame-ters vanishing or negative values of d2W are found but no failure was observed (after
a stress probe the numerical sample reach a new equilibrium state).

We verify hereafter that the control mode plays a fundamental role in failure
oc-currence. The objective is to impose a given stress direction to numerical samples
through different control parameters than previous one. The ratio between σ_{1} and
σ_{3} can be imposed for instance through the condition dσ_{1} − dσ3/R = 0, where
R = cos α /√2 sin αfor α ∈ ]180 deg; 270 deg]. By restricting our analysis to
con-jugated parameters in the sense of energy (Nova, 1994), then:

σ_{1}ε_{1} + 2 σ_{3}ε_{3} = ε_{1}
σ_{1} − σ3
R
+ (ε_{1} + 2R ε_{3}) σ3
R [5]

Thus the loading programme can be deﬁned for instance by:
dσ_{1} − dσ3

R = 0 and dε1 + 2R dε3 < 0 [6]

and the response parameters are dε_{1} and dσ_{3}/R. This loading programme ensures
a mixed control mode (in stress and strain) with control parameters deﬁned as linear

combinations of principal stress or strain components. Such control modes are
en-countered in very classical laboratory test, for instance a drained triaxial compression
corresponds to a mixed control mode (σ_{3} = 0 and dε_{1} > 0), and for an undrained
triaxial compression a condition is imposed on a linear combination of principal strain
components (dε_{v} = dε_{1} + 2 dε_{3} = 0).

Numerically, to impose the loading programme deﬁned by [6] to the sample,
sev-eral time steps are necessary (even if increments of control parameters are “small”).
At each time step t of the discrete element method (DEM) the following conditions
are veriﬁed:
C_{σ}t = σ_{1}t − σ
t
3
R and C
t
ε = εt1 + 2R εt3 [7]

where C_{σ}t and C_{ε}t are constants set at each step t such that conditions deﬁned in [6]
are veriﬁed when the simulation ends; σ_{1}t, σ_{3}t, εt_{1} and εt_{3} correspond to the stress-strain
state computed at step t. Figure 8 presents a simpliﬁed diagram of the numerical
loop followed, involving conditions deﬁned in [7]. Full details can be found in Sibille
(2006).

**Figure 8. Principle of the loop to control linear combinations of principal stress or***strain components*

In a ﬁrst time we consider the stress direction characterised by R = 1
(α = 215.3 deg). This direction is included in the cone of unstable stress directions
for sample E3 at the stress state considered (Figure 7). For R = 1 control parameters
deﬁned in [6] write: dq = 0 and dε_{v} < 0 (dilatancy imposed). Thus the sample
fol-lows a constant shear stress path (for instance, such a stress path can be followed by

a material point in a slope when pore-water pressure increases: dq = 0 and dp < 0).
Figure 9 shows a comparison of responses of E3 between a full stress control and
a control deﬁned by [6] in terms of kinetic energy and stresses. The kinetic energy
of the sample is computed as the sum of the kinetic energy of each grain. When the
sample is fully stress controlled the kinetic energy stays very low and ﬁnally vanishes,
the sample reaches a new equilibrium at a stress state close to the initial one. When
the sample is controlled by dq = 0 and dε_{v} < 0a burst of kinetic energy is computed
characterised by a monotonic increase until values of two order of magnitude larger
than maximum kinetic energy computed during fully stress controlled probes. We
ob-serve a sudden vanishing of stresses (as a sudden liquefaction). Due to the dynamic
response, the importance of inertial terms can by visualized in the evolution of the
axial σ_{1} and radial σ_{3} stresses; when σ_{3} vanishes, σ_{1} is about 40 kPa.

This result veriﬁes that failure occurrence from a bifurcation point, detected with
the second-order work criterion, depends on the choice of the control parameters. For
instance, when the sample is fully stress controlled no failure can occur before the
*plastic limit condition as shown by Nova (1994). Darve et al. (2004) have shown*
that generalized limit states exist strictly inside the plastic limit condition. For the
particular case of constant shear stress paths on loose sand, the authors have shown that
a maximum dilatancy cannot be exceeded (as a stress state corresponding to the plastic
limit condition cannot be exceeded). The choice of the loading condition dε_{v} < 0
has been motivated by this demonstration. We verify, for this very particular loading
programme, that effectively a generalized limit state cannot be exceeded (note that if
dε_{v} > 0 was imposed no failure would be observed, in the same way as a “stress
unloading” applied from a plastic limit state does not lead to the failure).

In the following, we pay attention to the importance of the stress direction, to-gether with the inﬂuence of density, on failure occurrence. Four stress directions are considered and their belonging to cones of unstable stress directions for sample E1 and E3 (see Figure 7) are summarized in Table 2. The control parameters are those deﬁned in expression [6], but no variation of loading parameters is imposed. Thus the loading programme is deﬁned by:

dσ_{1} − dσ3

R = 0 and dε1 + 2R dε3 = 0 [8]

and apply to numerical samples through the algorithm presented in Figure 8.

If simulations are run in these conditions there is no evolution of the samples, they stay at their initial mechanical state governed by the prescribed control parameters. A perturbation is necessary to conclude about the importance of the stress direction. Simulations are performed without gravity. Therefore, at a given mechanical state, some spheres ﬂoat within the pores. Samples are perturbed by imposing, at a given time step t, an instantaneous velocity in a random direction to eight ﬂoating grains. Samples are virtually split into eight sub-parallelepipeds, each perturbed ﬂoating grain is chosen randomly in each sub-parallelepiped respectively. The velocity imposed to each grain is computed such that the value of kinetic energy provided is equal for each grain. The total value of external input of kinetic energy is 10−5 J, which is small with

respect to the maximum value of kinetic energy computed for fully stress controlled
probes: 10−4 _{J.}
0 0.02 0.04 0.06 0.08
0
0.002
0.004
0.006
0.008
0.01
Simulation time (s)
Kinetic energy (J)
total stress
control
control in
dq = 0 and d
ε < 0v
0 0.02 0.04 0.06 0.08
0
20
40
60
80
100
120
140
160
Simulation time (s)
σ 1
or
σ 3
(kPa)
σ_{3}
σ_{1}
total stress
control
control in
dq = 0
and
dε_{v} < 0

**Figure 9. Comparison of reponses of sample E3 along a constant shear stress loading**

*path control by dσ*_{1} = dσ_{3} *< 0 or by dq = 0 and dε*_{v} < 0

**Table 2. Stress directions and belonging to cones of unstable stress directions***(σ*_{3} *= 100 kPa)*

α R ∈ cone of unstable stress directions?

(deg) sample E1 (η = 0.82) sample E3 (η = 0.46)

200 1.94 no no

220 0.843 no yes

230 0.593 yes yes (close to the limit of the cone)

250 0.257 no no

Responses of samples E1 and E3 to the perturbation with respect to R value are
presented in Figures 10 and 11 in terms of kinetic energy and radial stress, σ_{3},

evo-lutions. For R = 1.94 and 0.257 (α = 200 and 250 deg respectively) samples E1 and E3 reach a new equilibrium at a stress state close to the initial one. These di-rections are not included in cones for both samples. For R = 0.843 (α = 220 deg) failure is observed only for sample E3, this direction is not included in the cone for sample E1. For R = 0.593 (α = 230 deg), failure is observed for both samples, this direction corresponds to negative values of d2W for both samples. However the burst of kinetic energy is not obvious for sample E3, for which direction α = 230 deg is close to the limit of the cone. These results verify that failure can occur only along loading directions included in the cone of unstable stress directions. These unstable stress directions are closely related to the density of the granular material.

α = 200 deg; R = 1.94 0 0.02 0.04 0.06 0.08 0 0.005 0.01 0.015 0.02 Simulation time (s) Kinetic energy (J) perturbation E1 E3 0 0.02 0.04 0.06 0.08 0 20 40 60 80 100 120 Simulation time (s) σ 3 (kPa) perturbation E3 E1 α = 220deg; R = 0.843 0 0.02 0.04 0.06 0.08 0 0.005 0.01 0.015 0.02 Simulation time (s) Kinetic energy (J) perturbation E3 E1 0 0.02 0.04 0.06 0.08 0 20 40 60 80 100 120 Simulation time (s) σ 3 (kPa) perturbation E3 E1

**Figure 10. Comparison of responses, to a perturbation in kinetic energy, computed***with samples E1 and E3, for α = 200 and 220 deg*

**5. Conclusion**

Since the discrete element method involves few hypotheses, it constitutes a good tool to verify prediction deduced from analytical developments. Moreover, for very particular loading programmes, as those deﬁned in this paper, simulations are easier to realize, than real experiments. The presented results conﬁrm that from bifurcation

points detected with the second-order work criterion, for stress directions included in
cones of unstable stress directions, and for particular control parameters, failure can
develop suddenly, characterised by a dynamic response of the material and the
van-ishing of stresses. These failures occur from stress states inside the Mohr-Coulomb
criterion and could explain, for instance, landslides not predicted by a classical
ap-proach.
α = 230deg; R = 0.593
0 0.02 0.04 0.06 0.08
0
0.005
0.01
0.015
0.02
Simulation time (s)
Kinetic energy (J)
perturbation E3
E1
0 0.02 0.04 0.06 0.08
0
20
40
60
80
100
120
Simulation time (s)
σ 3
(kPa)
perturbation
E3
E1
α = 250deg; R = 0.257
0 0.02 0.04 0.06 0.08
0
0.005
0.01
0.015
0.02
Simulation time (s)
Kinetic energy (J)
perturbation _{E3}
E1
0 0.02 0.04 0.06 0.08
0
20
40
60
80
100
120
Simulation time (s)
σ 3
(kPa)
perturbation
E3
E1

**Figure 11. Comparison of responses, to a perturbation in kinetic energy, computed***with samples E1 and E3, for α = 230 and 250 deg*

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